Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets
Introduction
In fuzzy set (FS) theory, which is a generalization of classical set theory introduced by Zadeh [36], entropy, distance measure and similarity measure have drawn the attention of many researchers who have studied these three concepts from different points of view. Similarity and distance measures have been widely applied in many fields such as multicriteria decision-making [5], [19], group decision [25], [31], grey relational analysis [24], pattern recognition [11], [13], image processing [14], and cluster analysis [35]. Since the similarity and distance measures of FSs and their extensions [19], [37] have been applied to many real-world situations, the entropy measure [1], [3], [10] can be naturally applied to such fields due to its close relationship with similarity measure and distance measure. In recent years, Torra [20] introduced the concept of hesitant fuzzy set (HFS) as an extension of FS in which the membership degree of a given element, called the hesitant fuzzy element (HFE), is defined as a set of possible values. This situation can be found in a group decision making problem. To clarify the necessity of introducing HFSs, consider a situation in which two decision makers discuss the membership degree of an element x to a set A, one wants to assign 0.2, but the other 0.4. Accordingly, the difficulty of establishing a common membership degree is not because there is a margin of error (as in intuitionistic fuzzy sets), or some possibility distribution values (as in type-2 fuzzy sets), but because there is a set of possible values. HFSs can be applied in many decision making problems (see [23], [27], [42]). Later, a number of other extensions of the HFSs have been developed such as dual hesitant fuzzy sets (DHFSs) [41], generalized hesitant fuzzy sets (GHFSs) [15] and hesitant fuzzy linguistic term sets (HFLTSs) [16].
A growing number of studies focus on the distance measure, the similarity measure and the entropy for the concepts of FSs [36], intuitionistic fuzzy sets (IFSs) [1] and interval-valued intuitionistic fuzzy sets (IVIFSs) [23], [24]. Distance measures are fundamentally important in various fields such as decision making, market prediction, and pattern recognition. The most widely used distance measures for FSs [4], [12], IFSs and IVIFSs [7], [28], [30] are the Euclidean distance, the Hamming distance and the Hausdorff metric. Moreover, a number of other extensions of the latter distance measures have been proposed for HFSs [33] and linguistic fuzzy sets [29]. The notion of entropy of IFSs was first introduced by Burillo and Bustince [2]. A non-probabilistic-type entropy measure with a geometric interpretation of IFSs was then proposed by Szmidt and Kacprzyk [17]. Exploiting the concept of probability, Hung and Yang [10] gave axiomatic definitions of entropy for IVFSs and IFSs. Furthermore, other scholars have introduced various entropy formulas for IVFSs [38], [39], [40] and IFSs [21], [22]. The notion of similarity measure of IFSs indicates the similarity degree of IFSs and has been widely applied in many fuzzy decision-making problems. Lots of studies have been done on this issue, for instance, Szmidt and Kacprzyk [18], Hong and Kim [8], Hung and Yang [9] and Xu and Chen [30] independently proposed various similarity measures based on different distance measures for IFSs.
However, few studies [34] focused on the relationship between the entropy, the similarity measure and the distance measure for HFSs, in particular, on the systematic transformation of the entropy into the similarity measure for HFSs and vice versa. Using this transformation, we can derive more formulas for the entropy and the similarity measure of HFSs. To the best of our knowledge, such transformations have not been established, even for HFEs [34]. Moreover, hesitant fuzzy entropy and cross-entropy introduced by Xu and Xia [34] are defined only for HFEs, not for HFSs. With counterexamples, it is shown that the entropies proposed by Xu and Xia [34] cannot discriminate some HFEs, even though they are apparently different. To overcome this difficulty, new entropies are proposed here and then the relationship between the entropy, the similarity measure and the distance measure for HFSs is studied. In this paper, at first a new axiomatic definition of entropy for HFSs is introduced and sets of entropies and similarity measures for HFSs induced by the hesitant fuzzy distance are then proposed.
The rest of the paper is organized as follows. In Section 2, we review some definitions and axioms to describe the information measures for HFSs, which will be used in the analysis throughout this paper. Section 3 is devoted to the main results concerning the transformation of the information measures for HFSs. Furthermore, in Section 3, the actual need of proposing a new entropy for HFSs is shown. Similar to the main results obtained for HFSs, we can also obtain analogous results for IVHFSs in Section 4. In Section 5, two clustering algorithms under hesitant fuzzy environment are developed in which the similarity measures of HFSs and IVHFSs are the applied indices in data analysis and classification. Moreover, two practical examples are provided to compare the proposed methods with the existing ones. This paper is concluded in Section 6.
Section snippets
Information measures for hesitant fuzzy sets
This section starts with the definition of hesitant fuzzy sets (HFSs), which were first introduced by Torra [20]. Moreover, we describe the axiomatic definitions of information measures, namely, distance measure, similarity measure and entropy between HFSs, which are used later to discuss the clustering algorithm based on HFSs.
Throughout this paper, X = {x1, x2, … , xn} is used frequently to denote the discourse set. Definition 2.1 A hesitant fuzzy set (HFS) M on X is defined in terms of a function hM(x) when[20]
Relationship between information measures for HFSs
Throughout this section, we investigate the relationship between the distance, the similarity measure and the entropy for HFSs based on their axiomatic definitions. In order to put forward some new formulas, a sequence of theorems are provided on how the mentioned information measures for HFSs can be transformed by each other.
Information measures for interval-valued hesitant fuzzy sets
In hesitant fuzzy decision making problems, the experts assign exact values for the membership of x in a HFS M. In reality, the membership degrees of a certain element x to M are not necessarily real numbers. They may be a range of values belonging to [0, 1]. To deal with such cases, Chen et al. [6] introduced the concept of interval-valued hesitant fuzzy set (IVHFS), which is a generalization of HFS. This generalization is similar to that encountered in intuitionistic fuzzy environments, where
Clustering algorithms for HFSs and IVHFSs
Inspired by the hesitant fuzzy clustering algorithm proposed by Chen et al. [6], which involves the correlation coefficient of HFSs, we first develop an algorithm to do clustering under hesitant fuzzy environments by the use of the similarity measure of HFSs. Then, this algorithm is extended to the case of IVHFSs. Definition 5.1 Let Aj (j = 1, 2, … , m) be m HFSs, and C = [ξij]m×m be a similarity matrix, where ξij = S(Ai, Aj) denotes the similarity measure of two HFSs Ai and Aj and satisfies: (1) 0 ⩽ ξij ⩽ 1, i, j = 1, 2, … , m;
Conclusion
In this paper, a mutual transformation of the entropy into the similarity measure was introduced for both HFSs and IVHFSs. By the experimental results, it was shown that the proposed entropies for HFSs produce better results than the existing ones. Two illustrative examples, namely HFS and IVHFS clustering problems, were given to show the validity and applicability of the proposed information measures. Since the similarity measure and the distance measure of HFSs and IVHFSs are very much
Acknowledgments
The author thanks the valuable suggestions from the three anonymous referees and Professor Witold Pedrycz, the editor in chief, for improving the quality of the paper. The author is also very grateful to Professor Dr. M. Xia for her insightful and constructive comments that led to an improved version of this paper.
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